Flow Simulation around Shapes (Includes Free Code)

     This post is about the simulation of flow 🍃 around and through various objects 🪈 and obstacles ⭕. The simulated cases include flow through a partially blocked pipe, shown in Fig. 1. Flow around flat plates arranged in the shape of the letter "T", shown in Fig. 2. Flow around a wedge and flow around a triangle 🔺, which are shown in Fig. 3 and 4 respectively.

     To create obstacles in Python 🐍, the Path statement is used. This is similar to the inpolygon statement in MATLAB 🧮. For example, the following code 🖳 is used to create an ellipse ⬭.

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid

theta = np.linspace(0, np.pi, 100) # create equally spaced angles from 0 to pi

a = 0.05 # semi-major axis (along x)

b = 0.025 # semi-minor axis (along y)

shape_center_x = L / 2 # shape center x

shape_center_y = 0 # ellipse center x

x1 = shape_center_x + (a * np.cos(theta)) # x-coordinates

y1 = shape_center_y + (b * np.sin(theta)) # y-coordinates

shape_path = Path(np.column_stack((x1, y1))) # shape region

points = np.vstack((X.ravel(), Y.ravel())).T # mark inside region

custom_curve = shape_path.contains_points(points).reshape(X.shape) # create boolean mask

custom_curve_boundary = np.zeros_like(custom_curve, dtype=bool) # find boundary of shape

for i in range(1, Nx-1):

    for j in range(1, Ny-1):

        if custom_curve.T[i, j]: # inside shape

            if (not custom_curve.T[i+1, j] or not custom_curve.T[i-1, j] or 

                not custom_curve.T[i, j+1] or not custom_curve.T[i, j-1]):

                custom_curve_boundary.T[i, j] = True # mark as boundary

boundary_indices = np.where(custom_curve_boundary.T) # mark boundary

right_neighbors = (boundary_indices[0] + 1, boundary_indices[1]) # right index

left_neighbors = (boundary_indices[0] - 1, boundary_indices[1]) # left index

top_neighbors = (boundary_indices[0], boundary_indices[1] + 1) # top index

bottom_neighbors = (boundary_indices[0], boundary_indices[1] - 1) # bottom index

valid_right = ~custom_curve.T[right_neighbors] # check if points are on shape boundary

valid_left = ~custom_curve.T[left_neighbors]

valid_top = ~custom_curve.T[top_neighbors]

valid_bottom = ~custom_curve.T[bottom_neighbors]

     The arrays x1 and y1 are created using the equation for ellipse with the required parameters. The ellipse region is marked using the Path statement. Indices inside the ellipse are marked using a Boolean mask. Nested loops are used to mark the ellipse boundary using a curve. The right, left, top and bottom neighbor points are identified to be used for the application of Neumann boundary conditions for pressure (no-slip). Within the time loop, "where" statement is used to identify and apply the Neumann boundary conditions on the ellipse wall. The same method applied on any shape, for example aero-foils, circles, nozzles etc. The complete code to reproduce Fig. 1 is made available 😇. 

     It should be noted that, that the mesh is still stairstep. It doesn't matter how small the mesh resolution 😲. This code is developed for educational and research purposes only as there is not much application to stairstep mesh in real world❗

Code

#Copyright <2025> <FAHAD BUTT>

#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

#%% import libraries

import numpy as np

import matplotlib.pyplot as plt

from matplotlib.path import Path

#%% define parameters

l_square = 1 # length of square

h = 0.025 / 25 # grid spacing

dt = 0.00001 # time step

L = 0.3 # domain length

D = 0.05 # domain depth

Nx = round(L / h) + 1 # grid points in x-axis

Ny = round(D / h) + 1 # grid points in y-axis

nu = 1 / 100 # kinematic viscosity

Uinf = 1 # free stream velocity / inlet velocity / lid velocity

cfl = dt * Uinf / h # cfl number

travel = 5 # times the disturbance travels entire length of computational domain

TT = travel * L / Uinf # total time

ns = int(TT / dt) # number of time steps

Re = round(l_square * Uinf / nu) # Reynolds number

#%% initialize variables

u = np.zeros((Nx, Ny)) # x-velocity

v = np.zeros((Nx, Ny)) # y-velocity

p = np.zeros((Nx, Ny)) # pressure

#%% create a shape

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid

theta = np.linspace(0, np.pi, 100) # create equally spaced angles from 0 to pi

a = 0.05 # semi-major axis (along x)

b = 0.025 # semi-minor axis (along y)

shape_center_x = L / 2 # shape center x

shape_center_y = 0 # ellipse center x

x1 = shape_center_x + (a * np.cos(theta)) # x-coordinates

y1 = shape_center_y + (b * np.sin(theta)) # y-coordinates

shape_path = Path(np.column_stack((x1, y1))) # shape region

points = np.vstack((X.ravel(), Y.ravel())).T # mark inside region

custom_curve = shape_path.contains_points(points).reshape(X.shape) # create boolean mask

custom_curve_boundary = np.zeros_like(custom_curve, dtype=bool) # find boundary of shape

for i in range(1, Nx-1):

    for j in range(1, Ny-1):

        if custom_curve.T[i, j]: # inside shape

            if (not custom_curve.T[i+1, j] or not custom_curve.T[i-1, j] or 

                not custom_curve.T[i, j+1] or not custom_curve.T[i, j-1]):

                custom_curve_boundary.T[i, j] = True # mark as boundary

boundary_indices = np.where(custom_curve_boundary.T) # mark boundary

right_neighbors = (boundary_indices[0] + 1, boundary_indices[1]) # right index

left_neighbors = (boundary_indices[0] - 1, boundary_indices[1]) # left index

top_neighbors = (boundary_indices[0], boundary_indices[1] + 1) # top index

bottom_neighbors = (boundary_indices[0], boundary_indices[1] - 1) # bottom index

valid_right = ~custom_curve.T[right_neighbors] # check if points are on shape boundary

valid_left = ~custom_curve.T[left_neighbors]

valid_top = ~custom_curve.T[top_neighbors]

valid_bottom = ~custom_curve.T[bottom_neighbors]

#%% solve 2D Navier-Stokes equations

for nt in range(ns):

    pn = p.copy()

    p[1:-1, 1:-1] = (pn[2:, 1:-1] + pn[:-2, 1:-1] + pn[1:-1, 2:] + pn[1:-1, :-2]) / 4 - h / (8 * dt) * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure

    # apply pressure boundary conditions

    p[0, :] = p[1, :] # dp/dx = 0 at x = 0

    p[-1, :] = 0 # p = 0 at x = L

    p[:, 0] = p[:, 1] # dp/dy = 0 at y = 0

    p[:, -1] = p[:, -2] # dp/dy = 0 at y = D / 2

    p[custom_curve.T] = 0 # p = 0 inside shape

    # dp/dn = 0 at shape boundary (no slip)

    p[boundary_indices] = np.where(valid_right, p[right_neighbors], p[boundary_indices]) # right neighbor is fluid

    p[boundary_indices] = np.where(valid_left, p[left_neighbors], p[boundary_indices]) # left neighbor is fluid

    p[boundary_indices] = np.where(valid_top, p[top_neighbors], p[boundary_indices]) # top neighbor is fluid

    p[boundary_indices] = np.where(valid_bottom, p[bottom_neighbors], p[boundary_indices]) # bottom neighbor is fluid

    un = u.copy()

    vn = v.copy()

    u[1:-1, 1:-1] = (un[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (un[2:, 1:-1] - un[:-2, 1:-1]) + vn[1:-1, 1:-1] * (un[1:-1, 2:] - un[1:-1, :-2])) - dt / (2 * h) * (p[2:, 1:-1] - p[:-2, 1:-1]) + (1 / Re) * dt / h**2 * (un[2:, 1:-1] + un[:-2, 1:-1] + un[1:-1, 2:] + un[1:-1, :-2] - 4 * un[1:-1, 1:-1])) # x momentum

    # u boundary conditions

    u[0, :] = Uinf # u = Uinf at x = 0

    u[-1, :] = u[-2, :] # du/dx = 0 at x = L

    u[:, 0] = 0 # u = 0 at y = 0

    u[:, -1] = u[:, -2] # du/dy = 0 at y = D / 2

    u[custom_curve.T] = 0 # u = 0 inside shape

    u[custom_curve_boundary.T] = 0 # no slip

    v[1:-1, 1:-1] = (vn[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (vn[2:, 1:-1] - vn[:-2, 1:-1]) + vn[1:-1, 1:-1] * (vn[1:-1, 2:] - vn[1:-1, :-2])) - dt / (2 * h) * (p[1:-1, 2:] - p[1:-1, :-2]) + (1 / Re) * dt / h**2 * (vn[2:, 1:-1] + vn[:-2, 1:-1] + vn[1:-1, 2:] + vn[1:-1, :-2] - 4 * vn[1:-1, 1:-1])) # y momentum

    # v boundary conditions

    v[0, :] = 0 # v = 0 at x = 0

    v[-1, :] = v[-2, :] # dv/dx = 0 at x = L

    v[:, 0] = 0 # v = 0 at y = 0

    v[:, -1] = 0 # v = 0 at y = D / 2

    v[custom_curve.T] = 0 # u = 0 inside shape

    v[custom_curve_boundary.T] = 0 # no slip

#%% post process

u1 = u.copy() # u-velocity for plotting with shape

v1 = v.copy() # v-velocity for plotting with shape

p1 = p.copy() # pressure for plotting with shape

# shape geometry for plotting

u1[custom_curve.T] = np.nan

v1[custom_curve.T] = np.nan

p1[custom_curve.T] = np.nan

velocity_magnitude1 = np.sqrt(u1**2 + v1**2) # velocity magnitude with shape

# visualize velocity vectors and pressure contours

plt.figure(dpi = 500)

plt.contourf(X, Y, u1.T, 128, cmap = 'jet')

plt.plot(x1, y1, color='black', alpha = 1, linewidth = 2)

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.axis('off')

plt.show()

plt.figure(dpi = 500)

plt.contourf(X, Y, v1.T, 128, cmap = 'jet')

plt.plot(x1, y1, color='black', alpha = 1, linewidth = 2)

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.axis('off')

plt.show()

plt.figure(dpi = 500)

plt.contourf(X, Y, p1.T, 128, cmap = 'jet')

plt.plot(x1, y1, color='black', alpha = 1, linewidth = 2)

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.axis('off')

plt.show()

plt.figure(dpi = 500)

plt.streamplot(X, Y, u1.T, v1.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5, arrowstyle='->', arrowsize = 1)  # plot streamlines

plt.plot(x1, y1, color='black', alpha = 1, linewidth = 2)

plt.gca().set_aspect('equal', adjustable = 'box')

plt.axis('off')

plt.show()

Fig. 1, Flow in a clogged pipe

Fig. 2, Flow around a blunt obstacle

Fig. 3, Flow around asymmetric wedge

Fig. 4, Flow around a triangle

     Within Figs. 1 - 4, top row shows u and v components of velocity, bottom row shows pressure and streamlines 💫. 

     Thank you for reading! If you want to hire me as your next shinning post-doc, do let reach out!

CFD Wizardry: A 50-Line Python Marvel

     In abundant spare time ⏳, yours truly has implemented the non-conservative and non-dimensional form of the discretized Navier-Stokes 🍃 equations. The code 🖳 in it's simplest form is less than 50 lines including importing libraries and plotting! 😲 For validation, refer here. More examples and free code is available here, here and here. Happy codding!

Code

# Copyright <2025> <FAHAD BUTT>

# Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

# The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

# THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

import numpy as np

import matplotlib.pyplot as plt

l_square = 1 # length of square

h = l_square / 500 # grid spacing

dt = 0.00002 # time step

L = 1 # domain length

D = 1 # domain depth

Nx = round(L / h) + 1 # grid points in x-axis

Ny = round(D / h) + 1 # grid points in y-axis

nu = 1 / 100 # kinematic viscosity

Uinf = 1 # free stream velocity / inlet velocity / lid velocity

cfl = dt * Uinf / h # cfl number

travel = 200 # times the disturbance travels entire length of computational domain

TT = travel * L / Uinf # total time

ns = int(TT / dt) # number of time steps

Re = round(l_square * Uinf / nu) # Reynolds number

u = np.zeros((Nx, Ny)) # x-velocity

v = np.zeros((Nx, Ny)) # y-velocity

p = np.zeros((Nx, Ny)) # pressure

for nt in range(ns): # solve 2D Navier-Stokes equations

    pn = p.copy()

    p[1:-1, 1:-1] = (pn[2:, 1:-1] + pn[:-2, 1:-1] + pn[1:-1, 2:] + pn[1:-1, :-2]) / 4 - h / (8 * dt) * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure

    p[0, :] = p[1, :] # dp/dx = 0 at x = 0

    p[-1, :] = p[-2, :] # dp/dx = 0 at x = L

    p[:, 0] = p[:, 1] # dp/dy = 0 at y = 0

    p[:, -1] = 0 # p = 0 at y = D

    un = u.copy()

    vn = v.copy()

    u[1:-1, 1:-1] = (un[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (un[2:, 1:-1] - un[:-2, 1:-1]) + vn[1:-1, 1:-1] * (un[1:-1, 2:] - un[1:-1, :-2])) - dt / (2 * h) * (p[2:, 1:-1] - p[:-2, 1:-1]) + (1 / Re) * dt / h**2 * (un[2:, 1:-1] + un[:-2, 1:-1] + un[1:-1, 2:] + un[1:-1, :-2] - 4 * un[1:-1, 1:-1])) # x momentum

    u[0, :] = 0 # u = 0 at x = 0

    u[-1, :] = 0 # u = 0 at x = L

    u[:, 0] = 0 # u = 0 at y = 0

    u[:, -1] = Uinf # u = Uinf at y = D

    v[1:-1, 1:-1] = (vn[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (vn[2:, 1:-1] - vn[:-2, 1:-1]) + vn[1:-1, 1:-1] * (vn[1:-1, 2:] - vn[1:-1, :-2])) - dt / (2 * h) * (p[1:-1, 2:] - p[1:-1, :-2]) + (1 / Re) * dt / h**2 * (vn[2:, 1:-1] + vn[:-2, 1:-1] + vn[1:-1, 2:] + vn[1:-1, :-2] - 4 * vn[1:-1, 1:-1])) # y momentum

    v[0, :] = 0 # v = 0 at x = 0

    v[-1, :] = 0 # v = 0 at x = L

    v[:, 0] = 0 # v = 0 at y = 0

    v[:, -1] = 0 # v = 0 at y = D

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid

plt.figure(dpi = 200)

plt.contourf(X, Y, v.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.streamplot(X, Y, u.T, v.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5, arrowstyle='->', arrowsize = 1) # plot streamlines

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

Lid-Driven Cavity

     The case of lid-driven cavity in the turbulent flow regime can now be solved in reasonable amount of time. The results are shown in Fig. 1. I stopped the code while the flow is still developing as you are reading a blog and not a Q1 journal. 😆 Within Fig. 1, streamlines, v and u component of velocity and pressure are shown going from left to right and top to bottom. At the center of Fig. 1, the velocity magnitude is superimposed. As this is DNS, the smallest spatial scale resolved is ~8e-3 m [8 mm]. While, the smallest time-scale ⌛ resolved is ~8e-4 s [0.8 ms].

Fig. 1, The results at Reynolds number of 10,000

Free-Jet

          The case of free jets in the turbulent flow regime can now be solved in reasonable amount of time. The results are shown in Fig. 2. I stopped the code while the flow is still developing. Once again, I remind you that you are reading a blog and not a Q1 journal. 😆 Within Fig. 2, streamlines and species are shown. As this is DNS, the smallest spatial scale resolved is ~0.02 m [2 cm]. While, the smallest time-scale ⌛ resolved is ~4e-4 s [0.4 ms]. The code for implementing species, in this case temperature using the energy equation is available on the previous post.

Fig. 2, Free jet at Reynolds number 10000

Heated Room

     The benchmark case of mixed convection in an open room in the turbulent flow regime can now be solved in reasonable amount of time as well. The results are shown in Fig. 3. I stopped the code while the flow field stopped showing any changes. 😆 As this is DNS, the smallest spatial scale resolved is ~0.0144 m [1.44 cm]. While, the smallest time-scale ⌛ resolved is ~1e-4 s [1 ms]. The code for implementing species, in this case temperature using the energy equation is available on the previous post. In the previous post, the momentum equation has no changes as the gravity vector is at 0 m/s2. For this example, Boussinesq assumption is used.

Fig. 3, Flow inside a heated room at Reynolds number of 5000

Backward - Facing Step (BFS)

     Another benchmark case of flow around a backwards facing step can now be solved in reasonable amount of time as well. The flow is fully turbulent. The results are shown in Fig. 4. I stopped the code while the flow field is still developing. 😆 As this is DNS, the smallest spatial scale resolved is ~0.01 m [1 cm]. While, the smallest time-scale ⌛ resolved is ~1e-4 s [1 ms]. As can be seen from Fig. 4, there are no abnormalities in the flow field.

Fig. 4, Flow around a backwards facing step at Reynolds number of 10000

PS: I fully understand, there is no such thing as 2D turbulence 🍃. Just don't kill the vibe please 💫.

Artificial Compressibility

     The artificial compressibility method is now implemented in the code. The output is flow inside the lid-driven cavity at Reynolds number 10,000. The smallest scale resolved is at 0.001 m and smallest time scale resolved is at 0.0001 s. This version of code seems to be more stable as compared to the one that uses pressure Poisson equation. The results are shown in Fig. 5.

Fig. 5, Look at all those secondary vortices 😚

#Copyright <2025> <FAHAD BUTT>

#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

#%% import libraries

import numpy as np

import matplotlib.pyplot as plt

#%% define parameters

l_cr = 1 # characteristic length

h = l_cr / 1000 # grid spacing

dt = 0.0001 # time step

L = 1 # domain length

D = 1 # domain depth

Nx = round(L / h) + 1 # grid points in x-axis

Ny = round(D / h) + 1 # grid points in y-axis

nu = 1 / 10000 # kinematic viscosity

Uinf = 1 # free stream velocity / inlet velocity / lid velocity

cfl = dt * Uinf / h # cfl number

travel = 20 # times the disturbance travels entire length of computational domain

TT = travel * L / Uinf # total time

ns = int(TT / dt) # number of time steps

Re = round(l_cr * Uinf / nu) # Reynolds number

#%% intialization

u = np.zeros((Nx, Ny)) # x-velocity

v = np.zeros((Nx, Ny)) # y-velocity

p = np.zeros((Nx, Ny)) # pressure

#%% pre calculate for speed

P1 = dt / h

P2 = (2 / Re) * dt / h**2

P3 = 1 - (4 * P2)

#%% solve 2D Navier-Stokes equations

for nt in range(ns):

    pn = p.copy()

    p[1:-1, 1:-1] = pn[1:-1, 1:-1] - P1 * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure

    p[0, :] = p[1, :] # dp/dx = 0 at x = 0

    p[-1, :] = p[-2, :] # dp/dx = 0 at x = L

    p[:, 0] = p[:, 1] # dp/dy = 0 at y = 0

    p[:, -1] = p[:, -2] # dp/dy = 0 at y = D

    un = u.copy()

    vn = v.copy()

    u[1:-1, 1:-1] = un[1:-1, 1:-1] * P3 - P1 * (un[1:-1, 1:-1] * (un[2:, 1:-1] - un[:-2, 1:-1]) + vn[1:-1, 1:-1] * (un[1:-1, 2:] - un[1:-1, :-2]) + p[2:, 1:-1] - p[:-2, 1:-1]) + P2 * (un[2:, 1:-1] + un[:-2, 1:-1] + un[1:-1, 2:] + un[1:-1, :-2]) # x momentum

    u[0, :] = 0 # u = 0 at x = 0

    u[-1, :] = 0 # u = 0 at x = L

    u[:, 0] = 0 # u = 0 at y = 0

    u[:, -1] = Uinf # u = Uinf at y = D

    v[1:-1, 1:-1] = vn[1:-1, 1:-1] * P3 - P1 * (un[1:-1, 1:-1] * (vn[2:, 1:-1] - vn[:-2, 1:-1]) + vn[1:-1, 1:-1] * (vn[1:-1, 2:] - vn[1:-1, :-2]) + p[1:-1, 2:] - p[1:-1, :-2]) + P2 * (vn[2:, 1:-1] + vn[:-2, 1:-1] + vn[1:-1, 2:] + vn[1:-1, :-2]) # y momentum

    v[0, :] = 0 # v = 0 at x = 0

    v[-1, :] = 0 # v = 0 at x = L

    v[:, 0] = 0 # v = 0 at y = 0

    v[:, -1] = 0 # v = 0 at y = D

#%% post processing

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid

plt.figure(dpi = 200)

plt.contourf(X, Y, u.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 200)

plt.contourf(X, Y, v.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 200)

plt.contourf(X, Y, p.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 200)

plt.streamplot(X, Y, u.T, v.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5, arrowstyle='->', arrowsize = 1) # plot streamlines

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

velocity_magnitude = np.sqrt(u**2 + v**2)  # calculate velocity magnitude

plt.figure(dpi = 200)

plt.contourf(X, Y, velocity_magnitude.T, 128, cmap = 'plasma_r') # plot contours

plt.streamplot(X, Y, u.T, v.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.1, arrowstyle='->', arrowsize = 0.5) # plot streamlines

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.axis("off")

plt.show()

Relaxation

     The relaxation parameter along with equations is now added! The "omega" parameter can be adjusted to over / under relax the simulation according to requirements! At Reynolds number of 5000, the code provides up to 4x speed in convergence with over-relaxation.

#Copyright <2025> <FAHAD BUTT>

#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

#%% import libraries

import numpy as np

import matplotlib.pyplot as plt

#%% define parameters

l_cr = 1 # characteristic length

h = l_cr / 600 # grid spacing

dt = 0.0001 # time step

L = 1 # domain length

D = 1 # domain depth

Nx = round(L / h) + 1 # grid points in x-axis

Ny = round(D / h) + 1 # grid points in y-axis

nu = 1 / 5000 # kinematic viscosity

Uinf = 1 # free stream velocity / inlet velocity / lid velocity

cfl = dt * Uinf / h # cfl number

travel = 5 # times the disturbance travels entire length of computational domain

TT = travel * L / Uinf # total time

ns = int(TT / dt) # number of time steps

Re = round(l_cr * Uinf / nu) # Reynolds number

#%% intialization

u = np.zeros((Nx, Ny)) # x-velocity

v = np.zeros((Nx, Ny)) # y-velocity

p = np.zeros((Nx, Ny)) # pressure

u_new = u.copy() # x-velocity (relaxation)

v_new = v.copy() # y-velocity (relaxation)

p_new = p.copy() # pressure (relaxation)

#%% pre calculate for speed

P1 = dt / h

P2 = (2 / Re) * dt / h**2

P3 = 1 - (4 * P2)

omega = 4 # relaxation parameter (omega < 5)

#%% solve 2D Navier-Stokes equations

for nt in range(ns):

    pn = p.copy()

    p_new[1:-1, 1:-1] = pn[1:-1, 1:-1] - P1 * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure

    p[1:-1, 1:-1] = (1 - omega) * pn[1:-1, 1:-1] + omega * p_new[1:-1, 1:-1] # relaxation

    p[0, :] = p[1, :] # dp/dx = 0 at x = 0

    p[-1, :] = p[-2, :] # dp/dx = 0 at x = L

    p[:, 0] = p[:, 1] # dp/dy = 0 at y = 0

    p[:, -1] = p[:, -2] # dp/dy = 0 at y = D

    un = u.copy()

    vn = v.copy()

    u_new[1:-1, 1:-1] = un[1:-1, 1:-1] * P3 - P1 * (un[1:-1, 1:-1] * (un[2:, 1:-1] - un[:-2, 1:-1]) + vn[1:-1, 1:-1] * (un[1:-1, 2:] - un[1:-1, :-2]) + p[2:, 1:-1] - p[:-2, 1:-1]) + P2 * (un[2:, 1:-1] + un[:-2, 1:-1] + un[1:-1, 2:] + un[1:-1, :-2]) # x momentum

    u[1:-1, 1:-1] = (1 - omega) * un[1:-1, 1:-1] + omega * u_new[1:-1, 1:-1] # relaxation

    u[0, :] = 0 # u = 0 at x = 0

    u[-1, :] = 0 # u = 0 at x = L

    u[:, 0] = 0 # u = 0 at y = 0

    u[:, -1] = Uinf # u = Uinf at y = D

    v_new[1:-1, 1:-1] = vn[1:-1, 1:-1] * P3 - P1 * (un[1:-1, 1:-1] * (vn[2:, 1:-1] - vn[:-2, 1:-1]) + vn[1:-1, 1:-1] * (vn[1:-1, 2:] - vn[1:-1, :-2]) + p[1:-1, 2:] - p[1:-1, :-2]) + P2 * (vn[2:, 1:-1] + vn[:-2, 1:-1] + vn[1:-1, 2:] + vn[1:-1, :-2]) # y momentum

    v[1:-1, 1:-1] = (1 - omega) * vn[1:-1, 1:-1] + omega * v_new[1:-1, 1:-1] # relaxation

    v[0, :] = 0 # v = 0 at x = 0

    v[-1, :] = 0 # v = 0 at x = L

    v[:, 0] = 0 # v = 0 at y = 0

    v[:, -1] = 0 # v = 0 at y = D

#%% post processing

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny)) # spatial grid

plt.figure(dpi = 200)

plt.contourf(X, Y, u.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 200)

plt.contourf(X, Y, v.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 200)

plt.contourf(X, Y, p.T, 128, cmap = 'jet') # plot contours

plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 200)

plt.streamplot(X, Y, u.T, v.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5, arrowstyle='->', arrowsize = 1) # plot streamlines

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

velocity_magnitude = np.sqrt(u**2 + v**2)  # calculate velocity magnitude

plt.figure(dpi = 200)

plt.contourf(X, Y, velocity_magnitude.T, 128, cmap = 'plasma_r') # plot contours

plt.streamplot(X, Y, u.T, v.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.1, arrowstyle='->', arrowsize = 0.5) # plot streamlines

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.axis("off")

plt.show()

     

     Thank you for reading! If you want to hire me as your next shinning post-doc, do let reach out!

2D Heat Equation Code (Finite Difference Method)

     After 🎉 successfully teaching 👨‍🏫 the neural network about the 🔥 heat / diffusion 💉 equation, yours truly thought it is the time ⏱️ to write a simple code 🖥️ for discretized domain as well. A code yours truly created in abundant spare time is mentioned in this blog 📖.

     A complex (sinusoidal) 🌊 boundary condition is implemented. Dirichlet and Neumann boundary conditions are also implemented. The code is vectorized ↗ so there is only one loop 😁.  CFL condition is implemented in the time-step calculation so time-step ⏳ is adjusted based on the mesh size automatically 😇.

     For simple geometries, traditional numerical methods are is still better than PINNs. 🧠

Code

#Copyright <2024> <FAHAD BUTT>

#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

import numpy as np

import matplotlib.pyplot as plt

# mesh parameters

L = 1 # length of plate

D = np.pi # width of plate

h = 1 / 50 # grid size

Nx = int((L / h) + 1) # grid points in x-axis

Ny = int((D / h) + 1) # grid points in y-axis

alpha = 1 # thermal diffusivity

dt = h**2 / (4 * alpha) # time step (based on CFL condition)

total_time = 1 # total time

nt = int(total_time / dt) # total time steps

# initialization

T = np.zeros((Nx, Ny)) # initial condition

T_new = np.zeros_like(T)

x = np.linspace(0, L, Nx)

y = np.linspace(0, D, Ny)

# solve 2D-transient heat equation

for n in range(nt):

    T_new[1:-1, 1:-1] = T[1:-1, 1:-1] + ((alpha * dt) / h**2) * (T[2:, 1:-1] + T[:-2, 1:-1] + T[1:-1, 2:] + T[1:-1, :-2] - 4 * T[1:-1, 1:-1])

    T[:, :] = T_new

    # apply boundary conditions

    T[0, :] = np.sin(2 * np.pi * y / D) # T = sin(y) at x = 0

    T[-1, :] = T[-2, :] # dT/dx = 0 at x = L

    T[:, 0] = 0 # T = 0 at y = 0

    T[:, -1] = 0 # T = 0 at y = D

# plotting

X, Y = np.meshgrid(x, y, indexing="ij")

plt.figure(dpi = 500)

plt.contourf(X, Y, T, levels = 64, cmap = "jet")

plt.gca().set_aspect('equal', adjustable = 'box')

plt.colorbar(label = "Temperature")

plt.title("Temperature Distribution")

plt.xlabel("x")

plt.ylabel("y")

plt.show()

     The code creates the output as shown in Fig. 1.

Fig. 1, Temperature distribution

     Thank you for reading! If you want to hire me as a post-doc researcher in the fields of thermo-fluids and / or fracture mechanics, do reach out!

14th Step of the 12 steps to Navier-Stokes 😑

     After the tremendous success of 13th Step (thanks to the two people who read it, don't understand why they even bothered? 😂) The 14th step now exists! This case is called the case of flow around an obstacle! Like a box. ⬜ This is an unofficial continuation to this. If I get it approved by Dr. Barba, then it will be official. The original series is in Python but I coded this in MATLAB without using many MATLAB specific functions so the code can be translated to other programing languages 🖧 quite easily 😊.

     In terms of validation, Strouhal number is at 0.145 [1-4] for flow around a square cylinder at Re 100. This code gives St of 0.141. 🤓 Fig. 1 shows results from the code.

Fig. 1, Post processing

Code

%% clear and close

close all

clear

clc

beep off % annoying beep off :)

%% define spatial and temporal grids

l_square = 1; % length of square

h = l_square/10; % grid spacing

dt = 1; % time step

L = 40; % cavity length

D = 15; % cavity depth

Nx = round((L/h)+1); % grid points in x-axis

Ny = round((D/h)+1); % grid points in y-axis

nu = 0.000015111; % kinematic viscosity

Uinf = 0.0015111; % free stream velocity / inlet velocity  / lid velocity

cfl = dt*Uinf/h; % % cfl number

travel = 10; % times the disturbance travels entire length of computational domain

TT = travel*L/Uinf; % total time

ns = TT/dt; % number of time steps

Re = l_square*Uinf/nu; % Reynolds number

rho = 1.2047; % fluid density

%% initialize flowfield

u = Uinf*ones(Nx,Ny); % x-velocity

v = zeros(Nx,Ny); % y-velocity

p = zeros(Nx,Ny); % pressure

i = 2:Nx-1; % spatial interior nodes in x-axis

j = 2:Ny-1; % spatial interior nodes in y-axis

[X, Y] = meshgrid(0:h:L, 0:h:D); % spatial grid

maxNumCompThreads('automatic'); % select CPU cores

%% solve 2D Navier-Stokes equations

for nt = 1:ns

    pn = p;

    p(i, j) = (pn(i+1, j)+pn(i-1, j)+pn(i, j+1)+pn(i, j-1))/4 ...

        -h*rho/(8*dt)*(u(i+1, j)-u(i-1, j)+v(i, j+1)-v(i, j-1)); % pressure poisson

    p(1, :) = p(2, :); % dp/dx = 0 at x = 0

    p(Nx, :) = 0; % p = 0 at x = L

    p(:, 1) = p(:, 2); % dp/dy = 0 at y = 0

    p(:, Ny) = p(:, Ny-1); % dp/dy = 0 at y = D

    p(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % box geometry

    p(round(5*Nx/L), round(7*Ny/D:8*Ny/D)) = p(round(5*Nx/L)-1, round(7*Ny/D:8*Ny/D)); % left side

    p(round(6*Nx/L), round(7*Ny/D:8*Ny/D)) = p(round(6*Nx/L)+1, round(7*Ny/D:8*Ny/D)); % right side

    p(round(5*Nx/L:6*Nx/L), round(7*Ny/D)) = p(round(5*Nx/L:6*Nx/L), round(7*Ny/D)-1); % bottom side

    p(round(5*Nx/L:6*Nx/L), round(8*Ny/D)) = p(round(5*Nx/L:6*Nx/L), round(8*Ny/D)+1); % top side

    un = u;

    vn = v;

    u(i, j) = un(i, j)-dt/(2 * h)*(un(i, j).*(un(i+1, j)-un(i-1, j))+vn(i, j).*(un(i, j+1)-un(i, j-1))) ...

        -dt/(2*rho*h)*(p(i+1, j)-p(i-1, j)) ...

        +nu*dt/h^2*(un(i+1, j)+un(i-1, j)+un(i, j+1)+un(i, j-1)-4*un(i, j)); % x-momentum

    u(1, :) = Uinf; % u = Uinf at x = L

    u(Nx, :) = u(Nx-1, :); % du/dx = 0 at x = L

    u(:, 1) = 0; % u = 0 at y = 0

    u(:, Ny) = 0; % u = 0 at y = D

    u(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % box geometry

    u(round(5*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % left side

    u(round(6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % right side

    u(round(5*Nx/L:6*Nx/L), round(7*Ny/D)) = 0; % bottom side

    u(round(5*Nx/L:6*Nx/L), round(8*Ny/D)) = 0; % top side

    v(i, j) = vn(i, j)-dt/(2*h)*(un(i, j).*(vn(i+1, j)-vn(i-1, j))+vn(i, j).*(vn(i, j+1)-vn(i, j-1))) ...

        -dt/(2*rho*h)*(p(i, j+1)-p(i, j-1)) ...

        + nu*dt/h^2*(vn(i+1, j)+vn(i-1, j)+vn(i, j+1)+vn(i, j-1)-4*vn(i, j)); % y-momentum

    v(1, :) = 0; % v = 0 at x = L

    v(Nx, :) = v(Nx-1, :); % dv/dx = 0 at x = L

    v(:, 1) = 0; % v = 0 at y = 0

    v(:, Ny) = 0; % v = 0 at y = D

    v(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % box geometry

    v(round(5*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % left side

    v(round(6*Nx/L), round(7*Ny/D:8*Ny/D)) = 0; % right side

    v(round(5*Nx/L:6*Nx/L), round(7*Ny/D)) = 0; % bottom side

    v(round(5*Nx/L:6*Nx/L), round(8*Ny/D)) = 0; % top side

end

%% post-processing

velocity_magnitude = sqrt(u.^2 + v.^2); % velocity magnitude

u1 = u; % u-velocity for plotting with box

v1 = v; % v-velocity for plotting with box

p1 = p; % p-velocity for plotting with box

u1(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = NaN; % step geometry

v1(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = NaN; % step geometry

p1(round(5*Nx/L:6*Nx/L), round(7*Ny/D:8*Ny/D)) = NaN; % step geometry

velocity_magnitude1 = sqrt(u1.^2 + v1.^2); % velocity magnitude with box

%% Visualize velocity vectors and pressure contours

hold on, axis off

contourf(X, Y, u1', 64, 'LineColor', 'none'); % contour plot

set(gca, 'FontSize', 20)

hh = streamslice(X, Y, u1', v1', 20); % streamlines

set(hh, 'Color', 'k','LineWidth', 01);

colorbar; % add color bar

colormap jet % set color map

axis equal % set true scale

xlim([0 L]); % set axis limits

ylim([2 13]);

xticks([0 L]) % set ticks

yticks([0 D]) % set ticks

xlabel('x [m]');

ylabel('y [m]');

Cite as:

Fahad Butt (2024). Flow Around Square Cylinder (https://fluiddynamicscomputer.blogspot.com/), Blogger. Retrieved Month Date, Year

Copyright <2024> <Fahad Butt>

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Python version:

     As the original series is in Python, here is the Python code for Step 14 of 12. Also, I removed mixed terms from pressure poisson equation, just because 😁.

# Copyright <2024> <FAHAD BUTT>

# Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

# The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

# THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

#%% import libraries

import numpy as np

import matplotlib.pyplot as plt

#%% define spatial and temporal grids

l_square = 1  # length of square

h = l_square / 10  # grid spacing

dt = 10  # time step

L = 35  # domain length

D = 21  # domain depth

Nx = round(L / h) + 1  # grid points in x-axis

Ny = round(D / h) + 1  # grid points in y-axis

nu = 0.000015111  # kinematic viscosity

Uinf = 0.0015111  # free stream velocity / inlet velocity / lid velocity

cfl = dt * Uinf / h  # cfl number

travel = 10 # times the disturbance travels entire length of computational domain

TT = travel * L / Uinf  # total time

ns = int(TT / dt)  # number of time steps

Re = round(l_square * Uinf / nu) # Reynolds number

rho = 1.2047  # fluid density

#%% initialize flowfield

u = Uinf * np.ones((Nx, Ny))  # x-velocity

v = np.zeros((Nx, Ny))  # y-velocity

p = np.zeros((Nx, Ny))  # pressure

X, Y = np.meshgrid(np.linspace(0, L, Nx), np.linspace(0, D, Ny))  # spatial grid

#%% Solve 2D Navier-Stokes equations

for nt in range(ns):

    pn = p.copy()

    p[1:-1, 1:-1] = (pn[2:, 1:-1] + pn[:-2, 1:-1] + pn[1:-1, 2:] + pn[1:-1, :-2]) / 4 - h * rho / (8 * dt) * (u[2:, 1:-1] - u[:-2, 1:-1] + v[1:-1, 2:] - v[1:-1, :-2]) # pressure

    # boundary conditions for pressure

    p[0, :] = p[1, :]  # dp/dx = 0 at x = 0

    p[-1, :] = 0  # p = 0 at x = L

    p[:, 0] = p[:, 1]  # dp/dy = 0 at y = 0

    p[:, -1] = p[:, -2]  # dp/dy = 0 at y = D

    p[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry

    p[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = p[round(10 * Nx / L) - 1, round(10 * Ny / D):round(11 * Ny / D)] # box left

    p[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = p[round(11 * Nx / L) + 1, round(10 * Ny / D):round(11 * Ny / D)] # box right

    p[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = p[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D) - 1] # box bottom

    p[round(10 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = p[round(10 * Nx / L):round(11 * Nx / L), round(11 * Ny / D) + 1] # box top

    un = u.copy()

    vn = v.copy()

    u[1:-1, 1:-1] = (un[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (un[2:, 1:-1] - un[:-2, 1:-1]) + vn[1:-1, 1:-1] * (un[1:-1, 2:] - un[1:-1, :-2])) - dt / (2 * rho * h) * (p[2:, 1:-1] - p[:-2, 1:-1]) + nu * dt / h**2 * (un[2:, 1:-1] + un[:-2, 1:-1] + un[1:-1, 2:] + un[1:-1, :-2] - 4 * un[1:-1, 1:-1])) # x momentum

    # boundary conditions for x-velocity

    u[0, :] = Uinf  # u = Uinf at x = L

    u[-1, :] = u[-2, :]  # du/dx = 0 at x = L

    u[:, 0] = Uinf  # u = 0 at y = 0

    u[:, -1] = Uinf  # u = 0 at y = D

    u[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry

    u[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box left

    u[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box right

    u[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = 0 # box bottom

    u[round(11 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = 0 # box top

    v[1:-1, 1:-1] = (vn[1:-1, 1:-1] - dt / (2 * h) * (un[1:-1, 1:-1] * (vn[2:, 1:-1] - vn[:-2, 1:-1]) + vn[1:-1, 1:-1] * (vn[1:-1, 2:] - vn[1:-1, :-2])) - dt / (2 * rho * h) * (p[1:-1, 2:] - p[1:-1, :-2]) + nu * dt / h**2 * (vn[2:, 1:-1] + vn[:-2, 1:-1] + vn[1:-1, 2:] + vn[1:-1, :-2] - 4 * vn[1:-1, 1:-1])) # y momentum

    # boundary conditions for y-velocity

    v[0, :] = 0  # v = 0 at x = L

    v[-1, :] = v[-2, :]  # dv/dx = 0 at x = L

    v[:, 0] = 0  # v = 0 at y = 0

    v[:, -1] = 0  # v = 0 at y = D

    v[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box geometry

    v[round(10 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box left

    v[round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = 0 # box right

    v[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D)] = 0 # box bottom

    v[round(11 * Nx / L):round(11 * Nx / L), round(11 * Ny / D)] = 0 # box top

#%% post-processing

velocity_magnitude = np.sqrt(u**2 + v**2)  # velocity magnitude

u1 = u.copy()  # u-velocity for plotting with box

v1 = v.copy()  # v-velocity for plotting with box

p1 = p.copy()  # p-velocity for plotting with box

# box geometry for plotting

u1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)] = np.nan

v1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)]  = np.nan

p1[round(10 * Nx / L):round(11 * Nx / L), round(10 * Ny / D):round(11 * Ny / D)]  = np.nan

velocity_magnitude1 = np.sqrt(u1**2 + v1**2)  # velocity magnitude with box

# visualize velocity vectors and pressure contours

plt.figure(dpi = 500)

plt.contourf(X, Y, u1.T, 128, cmap = 'hsv')

# plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 500)

plt.contourf(X, Y, v1.T, 128, cmap = 'turbo')

# plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

plt.figure(dpi = 500)

plt.contourf(X, Y, p1.T, 128, cmap = 'jet')

# plt.colorbar()

plt.gca().set_aspect('equal', adjustable='box')

plt.xticks([0, L])

plt.yticks([0, D])

plt.xlabel('x [m]')

plt.ylabel('y [m]')

plt.show()

References

[1] Khademinejad, Taha & Talebizadeh Sardari, Pouyan & Rahimzadeh, Hassan. (2015). Numerical Study of Unsteady Flow around a Square Cylinder in Compare with Circular Cylinder.

[2] Sohankar, A., Norbergb, C., Davidson, L., Numerical simulation of unsteady low-Reynolds number flow around rectangular cylinders at incidence, Journal of Wind Engineering and Industrial Aerodynamics, 69–71 (1997) 189-201.

[3] Cheng, M., Whyte, D. S., Lou, J., Numerical simulation of flow around a square cylinder in uniform-shear flow, Journal of Fluids and Structures, 23 (2007) 207–226.

[4] Lam, K., Lin, Y. F., L. Zou, Y. Liu, Numerical study of flow patterns and force characteristics for square and rectangular cylinders with wavy surfaces, Journal of Fluids and Structures, 28 (2012) 359–377.